| def semanticmodel::transformations::affine_matrix | ( | v, | ||
| q | ||||
| ) |
Create an affine transformation matrix from a 3d vector and a 4d quaternion
Definition at line 1875 of file transformations.py.
| def semanticmodel::transformations::affine_matrix_from_points | ( | v0, | ||
| v1, | ||||
shear = True, |
||||
scale = True, |
||||
usesvd = True | ||||
| ) |
Return affine transform matrix to register two point sets.
v0 and v1 are shape (ndims, \*) arrays of at least ndims non-homogeneous
coordinates, where ndims is the dimensionality of the coordinate space.
If shear is False, a similarity transformation matrix is returned.
If also scale is False, a rigid/Eucledian transformation matrix
is returned.
By default the algorithm by Hartley and Zissermann [15] is used.
If usesvd is True, similarity and Eucledian transformation matrices
are calculated by minimizing the weighted sum of squared deviations
(RMSD) according to the algorithm by Kabsch [8].
Otherwise, and if ndims is 3, the quaternion based algorithm by Horn [9]
is used, which is slower when using this Python implementation.
The returned matrix performs rotation, translation and uniform scaling
(if specified).
>>> v0 = [[0, 1031, 1031, 0], [0, 0, 1600, 1600]]
>>> v1 = [[675, 826, 826, 677], [55, 52, 281, 277]]
>>> affine_matrix_from_points(v0, v1)
array([[ 0.14549, 0.00062, 675.50008],
[ 0.00048, 0.14094, 53.24971],
[ 0. , 0. , 1. ]])
>>> T = translation_matrix(numpy.random.random(3)-0.5)
>>> R = random_rotation_matrix(numpy.random.random(3))
>>> S = scale_matrix(random.random())
>>> M = concatenate_matrices(T, R, S)
>>> v0 = (numpy.random.rand(4, 100) - 0.5) * 20
>>> v0[3] = 1
>>> v1 = numpy.dot(M, v0)
>>> v0[:3] += numpy.random.normal(0, 1e-8, 300).reshape(3, -1)
>>> M = affine_matrix_from_points(v0[:3], v1[:3])
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
More examples in superimposition_matrix()
Definition at line 879 of file transformations.py.
| def semanticmodel::transformations::angle_between_vectors | ( | v0, | ||
| v1, | ||||
directed = True, |
||||
axis = 0 | ||||
| ) |
Return angle between vectors. If directed is False, the input vectors are interpreted as undirected axes, i.e. the maximum angle is pi/2. >>> a = angle_between_vectors([1, -2, 3], [-1, 2, -3]) >>> numpy.allclose(a, math.pi) True >>> a = angle_between_vectors([1, -2, 3], [-1, 2, -3], directed=False) >>> numpy.allclose(a, 0) True >>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]] >>> v1 = [[3], [0], [0]] >>> a = angle_between_vectors(v0, v1) >>> numpy.allclose(a, [0, 1.5708, 1.5708, 0.95532]) True >>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]] >>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]] >>> a = angle_between_vectors(v0, v1, axis=1) >>> numpy.allclose(a, [1.5708, 1.5708, 1.5708, 0.95532]) True
Definition at line 1787 of file transformations.py.
| def semanticmodel::transformations::arcball_constrain_to_axis | ( | point, | ||
| axis | ||||
| ) |
Return sphere point perpendicular to axis.
Definition at line 1619 of file transformations.py.
| def semanticmodel::transformations::arcball_map_to_sphere | ( | point, | ||
| center, | ||||
| radius | ||||
| ) |
Return unit sphere coordinates from window coordinates.
Definition at line 1606 of file transformations.py.
| def semanticmodel::transformations::arcball_nearest_axis | ( | point, | ||
| axes | ||||
| ) |
Return axis, which arc is nearest to point.
Definition at line 1635 of file transformations.py.
| def semanticmodel::transformations::clip_matrix | ( | left, | ||
| right, | ||||
| bottom, | ||||
| top, | ||||
| near, | ||||
| far, | ||||
perspective = False | ||||
| ) |
Return matrix to obtain normalized device coordinates from frustrum. The frustrum bounds are axis-aligned along x (left, right), y (bottom, top) and z (near, far). Normalized device coordinates are in range [-1, 1] if coordinates are inside the frustrum. If perspective is True the frustrum is a truncated pyramid with the perspective point at origin and direction along z axis, otherwise an orthographic canonical view volume (a box). Homogeneous coordinates transformed by the perspective clip matrix need to be dehomogenized (devided by w coordinate). >>> frustrum = numpy.random.rand(6) >>> frustrum[1] += frustrum[0] >>> frustrum[3] += frustrum[2] >>> frustrum[5] += frustrum[4] >>> M = clip_matrix(perspective=False, *frustrum) >>> numpy.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1]) array([-1., -1., -1., 1.]) >>> numpy.dot(M, [frustrum[1], frustrum[3], frustrum[5], 1]) array([ 1., 1., 1., 1.]) >>> M = clip_matrix(perspective=True, *frustrum) >>> v = numpy.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1]) >>> v / v[3] array([-1., -1., -1., 1.]) >>> v = numpy.dot(M, [frustrum[1], frustrum[3], frustrum[4], 1]) >>> v / v[3] array([ 1., 1., -1., 1.])
Definition at line 586 of file transformations.py.
| def semanticmodel::transformations::compose_matrix | ( | scale = None, |
||
shear = None, |
||||
angles = None, |
||||
translate = None, |
||||
perspective = None | ||||
| ) |
Return transformation matrix from sequence of transformations.
This is the inverse of the decompose_matrix function.
Sequence of transformations:
scale : vector of 3 scaling factors
shear : list of shear factors for x-y, x-z, y-z axes
angles : list of Euler angles about static x, y, z axes
translate : translation vector along x, y, z axes
perspective : perspective partition of matrix
>>> scale = numpy.random.random(3) - 0.5
>>> shear = numpy.random.random(3) - 0.5
>>> angles = (numpy.random.random(3) - 0.5) * (2*math.pi)
>>> trans = numpy.random.random(3) - 0.5
>>> persp = numpy.random.random(4) - 0.5
>>> M0 = compose_matrix(scale, shear, angles, trans, persp)
>>> result = decompose_matrix(M0)
>>> M1 = compose_matrix(*result)
>>> is_same_transform(M0, M1)
True
Definition at line 799 of file transformations.py.
| def semanticmodel::transformations::concatenate_matrices | ( | matrices | ) |
Return concatenation of series of transformation matrices. >>> M = numpy.random.rand(16).reshape((4, 4)) - 0.5 >>> numpy.allclose(M, concatenate_matrices(M)) True >>> numpy.allclose(numpy.dot(M, M.T), concatenate_matrices(M, M.T)) True
Definition at line 1834 of file transformations.py.
| def semanticmodel::transformations::decompose_affine | ( | m | ) |
Decompose matrix into a rotation (represented as a quaternion) followed by a translation
Definition at line 1865 of file transformations.py.
| def semanticmodel::transformations::decompose_matrix | ( | matrix | ) |
Return sequence of transformations from transformation matrix.
matrix : array_like
Non-degenerative homogeneous transformation matrix
Return tuple of:
scale : vector of 3 scaling factors
shear : list of shear factors for x-y, x-z, y-z axes
angles : list of Euler angles about static x, y, z axes
translate : translation vector along x, y, z axes
perspective : perspective partition of matrix
Raise ValueError if matrix is of wrong type or degenerative.
>>> T0 = translation_matrix([1, 2, 3])
>>> scale, shear, angles, trans, persp = decompose_matrix(T0)
>>> T1 = translation_matrix(trans)
>>> numpy.allclose(T0, T1)
True
>>> S = scale_matrix(0.123)
>>> scale, shear, angles, trans, persp = decompose_matrix(S)
>>> scale[0]
0.123
>>> R0 = euler_matrix(1, 2, 3)
>>> scale, shear, angles, trans, persp = decompose_matrix(R0)
>>> R1 = euler_matrix(*angles)
>>> numpy.allclose(R0, R1)
True
Definition at line 714 of file transformations.py.
| def semanticmodel::transformations::euler_from_matrix | ( | matrix, | ||
axes = 'sxyz' | ||||
| ) |
Return Euler angles from rotation matrix for specified axis sequence. axes : One of 24 axis sequences as string or encoded tuple Note that many Euler angle triplets can describe one matrix. >>> R0 = euler_matrix(1, 2, 3, 'syxz') >>> al, be, ga = euler_from_matrix(R0, 'syxz') >>> R1 = euler_matrix(al, be, ga, 'syxz') >>> numpy.allclose(R0, R1) True >>> angles = (4*math.pi) * (numpy.random.random(3) - 0.5) >>> for axes in _AXES2TUPLE.keys(): ... R0 = euler_matrix(axes=axes, *angles) ... R1 = euler_matrix(axes=axes, *euler_from_matrix(R0, axes)) ... if not numpy.allclose(R0, R1): print(axes, "failed")
Definition at line 1102 of file transformations.py.
| def semanticmodel::transformations::euler_from_quaternion | ( | quaternion, | ||
axes = 'sxyz' | ||||
| ) |
Return Euler angles from quaternion for specified axis sequence. >>> angles = euler_from_quaternion([0.99810947, 0.06146124, 0, 0]) >>> numpy.allclose(angles, [0.123, 0, 0]) True
Definition at line 1160 of file transformations.py.
| def semanticmodel::transformations::euler_matrix | ( | ai, | ||
| aj, | ||||
| ak, | ||||
axes = 'sxyz' | ||||
| ) |
Return homogeneous rotation matrix from Euler angles and axis sequence. ai, aj, ak : Euler's roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple >>> R = euler_matrix(1, 2, 3, 'syxz') >>> numpy.allclose(numpy.sum(R[0]), -1.34786452) True >>> R = euler_matrix(1, 2, 3, (0, 1, 0, 1)) >>> numpy.allclose(numpy.sum(R[0]), -0.383436184) True >>> ai, aj, ak = (4*math.pi) * (numpy.random.random(3) - 0.5) >>> for axes in _AXES2TUPLE.keys(): ... R = euler_matrix(ai, aj, ak, axes) >>> for axes in _TUPLE2AXES.keys(): ... R = euler_matrix(ai, aj, ak, axes)
Definition at line 1039 of file transformations.py.
| def semanticmodel::transformations::identity_matrix | ( | ) |
Return 4x4 identity/unit matrix. >>> I = identity_matrix() >>> numpy.allclose(I, numpy.dot(I, I)) True >>> numpy.sum(I), numpy.trace(I) (4.0, 4.0) >>> numpy.allclose(I, numpy.identity(4)) True
Definition at line 197 of file transformations.py.
| def semanticmodel::transformations::inverse_matrix | ( | matrix | ) |
Return inverse of square transformation matrix. >>> M0 = random_rotation_matrix() >>> M1 = inverse_matrix(M0.T) >>> numpy.allclose(M1, numpy.linalg.inv(M0.T)) True >>> for size in range(1, 7): ... M0 = numpy.random.rand(size, size) ... M1 = inverse_matrix(M0) ... if not numpy.allclose(M1, numpy.linalg.inv(M0)): print(size)
Definition at line 1818 of file transformations.py.
| def semanticmodel::transformations::is_same_transform | ( | matrix0, | ||
| matrix1 | ||||
| ) |
Return True if two matrices perform same transformation. >>> is_same_transform(numpy.identity(4), numpy.identity(4)) True >>> is_same_transform(numpy.identity(4), random_rotation_matrix()) False
Definition at line 1850 of file transformations.py.
| def semanticmodel::transformations::orthogonalization_matrix | ( | lengths, | ||
| angles | ||||
| ) |
Return orthogonalization matrix for crystallographic cell coordinates. Angles are expected in degrees. The de-orthogonalization matrix is the inverse. >>> O = orthogonalization_matrix([10, 10, 10], [90, 90, 90]) >>> numpy.allclose(O[:3, :3], numpy.identity(3, float) * 10) True >>> O = orthogonalization_matrix([9.8, 12.0, 15.5], [87.2, 80.7, 69.7]) >>> numpy.allclose(numpy.sum(O), 43.063229) True
Definition at line 852 of file transformations.py.
| def semanticmodel::transformations::projection_from_matrix | ( | matrix, | ||
pseudo = False | ||||
| ) |
Return projection plane and perspective point from projection matrix. Return values are same as arguments for projection_matrix function: point, normal, direction, perspective, and pseudo. >>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> persp = numpy.random.random(3) - 0.5 >>> P0 = projection_matrix(point, normal) >>> result = projection_from_matrix(P0) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True >>> P0 = projection_matrix(point, normal, direct) >>> result = projection_from_matrix(P0) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True >>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=False) >>> result = projection_from_matrix(P0, pseudo=False) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True >>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=True) >>> result = projection_from_matrix(P0, pseudo=True) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True
Definition at line 513 of file transformations.py.
| def semanticmodel::transformations::projection_matrix | ( | point, | ||
| normal, | ||||
direction = None, |
||||
perspective = None, |
||||
pseudo = False | ||||
| ) |
Return matrix to project onto plane defined by point and normal. Using either perspective point, projection direction, or none of both. If pseudo is True, perspective projections will preserve relative depth such that Perspective = dot(Orthogonal, PseudoPerspective). >>> P = projection_matrix([0, 0, 0], [1, 0, 0]) >>> numpy.allclose(P[1:, 1:], numpy.identity(4)[1:, 1:]) True >>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> persp = numpy.random.random(3) - 0.5 >>> P0 = projection_matrix(point, normal) >>> P1 = projection_matrix(point, normal, direction=direct) >>> P2 = projection_matrix(point, normal, perspective=persp) >>> P3 = projection_matrix(point, normal, perspective=persp, pseudo=True) >>> is_same_transform(P2, numpy.dot(P0, P3)) True >>> P = projection_matrix([3, 0, 0], [1, 1, 0], [1, 0, 0]) >>> v0 = (numpy.random.rand(4, 5) - 0.5) * 20 >>> v0[3] = 1 >>> v1 = numpy.dot(P, v0) >>> numpy.allclose(v1[1], v0[1]) True >>> numpy.allclose(v1[0], 3-v1[1]) True
Definition at line 451 of file transformations.py.
| def semanticmodel::transformations::quaternion_about_axis | ( | angle, | ||
| axis | ||||
| ) |
Return quaternion for rotation about axis. >>> q = quaternion_about_axis(0.123, [1, 0, 0]) >>> numpy.allclose(q, [0.99810947, 0.06146124, 0, 0]) True
Definition at line 1228 of file transformations.py.
| def semanticmodel::transformations::quaternion_conjugate | ( | quaternion | ) |
Return conjugate of quaternion. >>> q0 = random_quaternion() >>> q1 = quaternion_conjugate(q0) >>> q1[0] == q0[0] and all(q1[1:] == -q0[1:]) True
Definition at line 1364 of file transformations.py.
| def semanticmodel::transformations::quaternion_from_euler | ( | ai, | ||
| aj, | ||||
| ak, | ||||
axes = 'sxyz' | ||||
| ) |
Return quaternion from Euler angles and axis sequence. ai, aj, ak : Euler's roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple >>> q = quaternion_from_euler(1, 2, 3, 'ryxz') >>> numpy.allclose(q, [0.435953, 0.310622, -0.718287, 0.444435]) True
Definition at line 1171 of file transformations.py.
| def semanticmodel::transformations::quaternion_from_matrix | ( | matrix, | ||
isprecise = False | ||||
| ) |
Return quaternion from rotation matrix. If isprecise is True, the input matrix is assumed to be a precise rotation matrix and a faster algorithm is used. >>> q = quaternion_from_matrix(numpy.identity(4), True) >>> numpy.allclose(q, [1, 0, 0, 0]) True >>> q = quaternion_from_matrix(numpy.diag([1, -1, -1, 1])) >>> numpy.allclose(q, [0, 1, 0, 0]) or numpy.allclose(q, [0, -1, 0, 0]) True >>> R = rotation_matrix(0.123, (1, 2, 3)) >>> q = quaternion_from_matrix(R, True) >>> numpy.allclose(q, [0.9981095, 0.0164262, 0.0328524, 0.0492786]) True >>> R = [[-0.545, 0.797, 0.260, 0], [0.733, 0.603, -0.313, 0], ... [-0.407, 0.021, -0.913, 0], [0, 0, 0, 1]] >>> q = quaternion_from_matrix(R) >>> numpy.allclose(q, [0.19069, 0.43736, 0.87485, -0.083611]) True >>> R = [[0.395, 0.362, 0.843, 0], [-0.626, 0.796, -0.056, 0], ... [-0.677, -0.498, 0.529, 0], [0, 0, 0, 1]] >>> q = quaternion_from_matrix(R) >>> numpy.allclose(q, [0.82336615, -0.13610694, 0.46344705, -0.29792603]) True >>> R = random_rotation_matrix() >>> q = quaternion_from_matrix(R) >>> is_same_transform(R, quaternion_matrix(q)) True
Definition at line 1271 of file transformations.py.
| def semanticmodel::transformations::quaternion_imag | ( | quaternion | ) |
Return imaginary part of quaternion. >>> quaternion_imag([3, 0, 1, 2]) array([ 0., 1., 2.])
Definition at line 1402 of file transformations.py.
| def semanticmodel::transformations::quaternion_inverse | ( | quaternion | ) |
Return inverse of quaternion. >>> q0 = random_quaternion() >>> q1 = quaternion_inverse(q0) >>> numpy.allclose(quaternion_multiply(q0, q1), [1, 0, 0, 0]) True
Definition at line 1378 of file transformations.py.
| def semanticmodel::transformations::quaternion_matrix | ( | quaternion | ) |
Return homogeneous rotation matrix from quaternion. >>> M = quaternion_matrix([0.99810947, 0.06146124, 0, 0]) >>> numpy.allclose(M, rotation_matrix(0.123, [1, 0, 0])) True >>> M = quaternion_matrix([1, 0, 0, 0]) >>> numpy.allclose(M, numpy.identity(4)) True >>> M = quaternion_matrix([0, 1, 0, 0]) >>> numpy.allclose(M, numpy.diag([1, -1, -1, 1])) True
Definition at line 1244 of file transformations.py.
| def semanticmodel::transformations::quaternion_multiply | ( | quaternion1, | ||
| quaternion0 | ||||
| ) |
Return multiplication of two quaternions. >>> q = quaternion_multiply([4, 1, -2, 3], [8, -5, 6, 7]) >>> numpy.allclose(q, [28, -44, -14, 48]) True
Definition at line 1348 of file transformations.py.
| def semanticmodel::transformations::quaternion_real | ( | quaternion | ) |
Return real part of quaternion. >>> quaternion_real([3, 0, 1, 2]) 3.0
Definition at line 1392 of file transformations.py.
| def semanticmodel::transformations::quaternion_slerp | ( | quat0, | ||
| quat1, | ||||
| fraction, | ||||
spin = 0, |
||||
shortestpath = True | ||||
| ) |
Return spherical linear interpolation between two quaternions.
>>> q0 = random_quaternion()
>>> q1 = random_quaternion()
>>> q = quaternion_slerp(q0, q1, 0)
>>> numpy.allclose(q, q0)
True
>>> q = quaternion_slerp(q0, q1, 1, 1)
>>> numpy.allclose(q, q1)
True
>>> q = quaternion_slerp(q0, q1, 0.5)
>>> angle = math.acos(numpy.dot(q0, q))
>>> numpy.allclose(2, math.acos(numpy.dot(q0, q1)) / angle) or \
numpy.allclose(2, math.acos(-numpy.dot(q0, q1)) / angle)
True
Definition at line 1412 of file transformations.py.
| def semanticmodel::transformations::random_quaternion | ( | rand = None |
) |
Return uniform random unit quaternion.
rand: array like or None
Three independent random variables that are uniformly distributed
between 0 and 1.
>>> q = random_quaternion()
>>> numpy.allclose(1, vector_norm(q))
True
>>> q = random_quaternion(numpy.random.random(3))
>>> len(q.shape), q.shape[0]==4
(1, True)
Definition at line 1453 of file transformations.py.
| def semanticmodel::transformations::random_rotation_matrix | ( | rand = None |
) |
Return uniform random rotation matrix.
rand: array like
Three independent random variables that are uniformly distributed
between 0 and 1 for each returned quaternion.
>>> R = random_rotation_matrix()
>>> numpy.allclose(numpy.dot(R.T, R), numpy.identity(4))
True
Definition at line 1481 of file transformations.py.
| def semanticmodel::transformations::random_vector | ( | size | ) |
Return array of random doubles in the half-open interval [0.0, 1.0). >>> v = random_vector(10000) >>> numpy.all(v >= 0) and numpy.all(v < 1) True >>> v0 = random_vector(10) >>> v1 = random_vector(10) >>> numpy.any(v0 == v1) False
Definition at line 1751 of file transformations.py.
| def semanticmodel::transformations::reflection_from_matrix | ( | matrix | ) |
Return mirror plane point and normal vector from reflection matrix. >>> v0 = numpy.random.random(3) - 0.5 >>> v1 = numpy.random.random(3) - 0.5 >>> M0 = reflection_matrix(v0, v1) >>> point, normal = reflection_from_matrix(M0) >>> M1 = reflection_matrix(point, normal) >>> is_same_transform(M0, M1) True
Definition at line 263 of file transformations.py.
| def semanticmodel::transformations::reflection_matrix | ( | point, | ||
| normal | ||||
| ) |
Return matrix to mirror at plane defined by point and normal vector. >>> v0 = numpy.random.random(4) - 0.5 >>> v0[3] = 1. >>> v1 = numpy.random.random(3) - 0.5 >>> R = reflection_matrix(v0, v1) >>> numpy.allclose(2, numpy.trace(R)) True >>> numpy.allclose(v0, numpy.dot(R, v0)) True >>> v2 = v0.copy() >>> v2[:3] += v1 >>> v3 = v0.copy() >>> v2[:3] -= v1 >>> numpy.allclose(v2, numpy.dot(R, v3)) True
Definition at line 237 of file transformations.py.
| def semanticmodel::transformations::rotation_from_matrix | ( | matrix | ) |
Return rotation angle and axis from rotation matrix. >>> angle = (random.random() - 0.5) * (2*math.pi) >>> direc = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> R0 = rotation_matrix(angle, direc, point) >>> angle, direc, point = rotation_from_matrix(R0) >>> R1 = rotation_matrix(angle, direc, point) >>> is_same_transform(R0, R1) True
Definition at line 336 of file transformations.py.
| def semanticmodel::transformations::rotation_matrix | ( | angle, | ||
| direction, | ||||
point = None | ||||
| ) |
Return matrix to rotate about axis defined by point and direction. >>> R = rotation_matrix(math.pi/2, [0, 0, 1], [1, 0, 0]) >>> numpy.allclose(numpy.dot(R, [0, 0, 0, 1]), [1, -1, 0, 1]) True >>> angle = (random.random() - 0.5) * (2*math.pi) >>> direc = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> R0 = rotation_matrix(angle, direc, point) >>> R1 = rotation_matrix(angle-2*math.pi, direc, point) >>> is_same_transform(R0, R1) True >>> R0 = rotation_matrix(angle, direc, point) >>> R1 = rotation_matrix(-angle, -direc, point) >>> is_same_transform(R0, R1) True >>> I = numpy.identity(4, numpy.float64) >>> numpy.allclose(I, rotation_matrix(math.pi*2, direc)) True >>> numpy.allclose(2, numpy.trace(rotation_matrix(math.pi/2, ... direc, point))) True
Definition at line 292 of file transformations.py.
| def semanticmodel::transformations::scale_from_matrix | ( | matrix | ) |
Return scaling factor, origin and direction from scaling matrix. >>> factor = random.random() * 10 - 5 >>> origin = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> S0 = scale_matrix(factor, origin) >>> factor, origin, direction = scale_from_matrix(S0) >>> S1 = scale_matrix(factor, origin, direction) >>> is_same_transform(S0, S1) True >>> S0 = scale_matrix(factor, origin, direct) >>> factor, origin, direction = scale_from_matrix(S0) >>> S1 = scale_matrix(factor, origin, direction) >>> is_same_transform(S0, S1) True
Definition at line 410 of file transformations.py.
| def semanticmodel::transformations::scale_matrix | ( | factor, | ||
origin = None, |
||||
direction = None | ||||
| ) |
Return matrix to scale by factor around origin in direction. Use factor -1 for point symmetry. >>> v = (numpy.random.rand(4, 5) - 0.5) * 20 >>> v[3] = 1 >>> S = scale_matrix(-1.234) >>> numpy.allclose(numpy.dot(S, v)[:3], -1.234*v[:3]) True >>> factor = random.random() * 10 - 5 >>> origin = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> S = scale_matrix(factor, origin) >>> S = scale_matrix(factor, origin, direct)
Definition at line 376 of file transformations.py.
| def semanticmodel::transformations::shear_from_matrix | ( | matrix | ) |
Return shear angle, direction and plane from shear matrix. >>> angle = (random.random() - 0.5) * 4*math.pi >>> direct = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.cross(direct, numpy.random.random(3)) >>> S0 = shear_matrix(angle, direct, point, normal) >>> angle, direct, point, normal = shear_from_matrix(S0) >>> S1 = shear_matrix(angle, direct, point, normal) >>> is_same_transform(S0, S1) True
Definition at line 669 of file transformations.py.
| def semanticmodel::transformations::shear_matrix | ( | angle, | ||
| direction, | ||||
| point, | ||||
| normal | ||||
| ) |
Return matrix to shear by angle along direction vector on shear plane. The shear plane is defined by a point and normal vector. The direction vector must be orthogonal to the plane's normal vector. A point P is transformed by the shear matrix into P2 such that the vector P-P2 is parallel to the direction vector and its extent is given by the angle of P-P'-P2, where P' is the orthogonal projection of P onto the shear plane. >>> angle = (random.random() - 0.5) * 4*math.pi >>> direct = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.cross(direct, numpy.random.random(3)) >>> S = shear_matrix(angle, direct, point, normal) >>> numpy.allclose(1, numpy.linalg.det(S)) True
Definition at line 638 of file transformations.py.
| def semanticmodel::transformations::superimposition_matrix | ( | v0, | ||
| v1, | ||||
scale = False, |
||||
usesvd = True | ||||
| ) |
Return matrix to transform given 3D point set into second point set. v0 and v1 are shape (3, \*) or (4, \*) arrays of at least 3 points. The parameters scale and usesvd are explained in the more general affine_matrix_from_points function. The returned matrix is a similarity or Eucledian transformation matrix. This function has a fast C implementation in transformations.c. >>> v0 = numpy.random.rand(3, 10) >>> M = superimposition_matrix(v0, v0) >>> numpy.allclose(M, numpy.identity(4)) True >>> R = random_rotation_matrix(numpy.random.random(3)) >>> v0 = [[1,0,0], [0,1,0], [0,0,1], [1,1,1]] >>> v1 = numpy.dot(R, v0) >>> M = superimposition_matrix(v0, v1) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> v0 = (numpy.random.rand(4, 100) - 0.5) * 20 >>> v0[3] = 1 >>> v1 = numpy.dot(R, v0) >>> M = superimposition_matrix(v0, v1) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> S = scale_matrix(random.random()) >>> T = translation_matrix(numpy.random.random(3)-0.5) >>> M = concatenate_matrices(T, R, S) >>> v1 = numpy.dot(M, v0) >>> v0[:3] += numpy.random.normal(0, 1e-9, 300).reshape(3, -1) >>> M = superimposition_matrix(v0, v1, scale=True) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> v = numpy.empty((4, 100, 3)) >>> v[:, :, 0] = v0 >>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False) >>> numpy.allclose(v1, numpy.dot(M, v[:, :, 0])) True
Definition at line 988 of file transformations.py.
| def semanticmodel::transformations::translation_from_matrix | ( | matrix | ) |
Return translation vector from translation matrix. >>> v0 = numpy.random.random(3) - 0.5 >>> v1 = translation_from_matrix(translation_matrix(v0)) >>> numpy.allclose(v0, v1) True
Definition at line 225 of file transformations.py.
| def semanticmodel::transformations::translation_matrix | ( | direction | ) |
Return matrix to translate by direction vector. >>> v = numpy.random.random(3) - 0.5 >>> numpy.allclose(v, translation_matrix(v)[:3, 3]) True
Definition at line 212 of file transformations.py.
| def semanticmodel::transformations::unit_vector | ( | data, | ||
axis = None, |
||||
out = None | ||||
| ) |
Return ndarray normalized by length, i.e. eucledian norm, along axis. >>> v0 = numpy.random.random(3) >>> v1 = unit_vector(v0) >>> numpy.allclose(v1, v0 / numpy.linalg.norm(v0)) True >>> v0 = numpy.random.rand(5, 4, 3) >>> v1 = unit_vector(v0, axis=-1) >>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=2)), 2) >>> numpy.allclose(v1, v2) True >>> v1 = unit_vector(v0, axis=1) >>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=1)), 1) >>> numpy.allclose(v1, v2) True >>> v1 = numpy.empty((5, 4, 3)) >>> unit_vector(v0, axis=1, out=v1) >>> numpy.allclose(v1, v2) True >>> list(unit_vector([])) [] >>> list(unit_vector([1])) [1.0]
Definition at line 1707 of file transformations.py.
| def semanticmodel::transformations::vector_norm | ( | data, | ||
axis = None, |
||||
out = None | ||||
| ) |
Return length, i.e. eucledian norm, of ndarray along axis. >>> v = numpy.random.random(3) >>> n = vector_norm(v) >>> numpy.allclose(n, numpy.linalg.norm(v)) True >>> v = numpy.random.rand(6, 5, 3) >>> n = vector_norm(v, axis=-1) >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=2))) True >>> n = vector_norm(v, axis=1) >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1))) True >>> v = numpy.random.rand(5, 4, 3) >>> n = numpy.empty((5, 3)) >>> vector_norm(v, axis=1, out=n) >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1))) True >>> vector_norm([]) 0.0 >>> vector_norm([1]) 1.0
Definition at line 1668 of file transformations.py.
| def semanticmodel::transformations::vector_product | ( | v0, | ||
| v1, | ||||
axis = 0 | ||||
| ) |
Return vector perpendicular to vectors. >>> v = vector_product([2, 0, 0], [0, 3, 0]) >>> numpy.allclose(v, [0, 0, 6]) True >>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]] >>> v1 = [[3], [0], [0]] >>> v = vector_product(v0, v1) >>> numpy.allclose(v, [[0, 0, 0, 0], [0, 0, 6, 6], [0, -6, 0, -6]]) True >>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]] >>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]] >>> v = vector_product(v0, v1, axis=1) >>> numpy.allclose(v, [[0, 0, 6], [0, -6, 0], [6, 0, 0], [0, -6, 6]]) True
Definition at line 1766 of file transformations.py.
| dictionary semanticmodel::transformations::_AXES2TUPLE |
{
'sxyz': (0, 0, 0, 0), 'sxyx': (0, 0, 1, 0), 'sxzy': (0, 1, 0, 0),
'sxzx': (0, 1, 1, 0), 'syzx': (1, 0, 0, 0), 'syzy': (1, 0, 1, 0),
'syxz': (1, 1, 0, 0), 'syxy': (1, 1, 1, 0), 'szxy': (2, 0, 0, 0),
'szxz': (2, 0, 1, 0), 'szyx': (2, 1, 0, 0), 'szyz': (2, 1, 1, 0),
'rzyx': (0, 0, 0, 1), 'rxyx': (0, 0, 1, 1), 'ryzx': (0, 1, 0, 1),
'rxzx': (0, 1, 1, 1), 'rxzy': (1, 0, 0, 1), 'ryzy': (1, 0, 1, 1),
'rzxy': (1, 1, 0, 1), 'ryxy': (1, 1, 1, 1), 'ryxz': (2, 0, 0, 1),
'rzxz': (2, 0, 1, 1), 'rxyz': (2, 1, 0, 1), 'rzyz': (2, 1, 1, 1)}
Definition at line 1655 of file transformations.py.
| tuple semanticmodel::transformations::_EPS = numpy.finfo(float) |
Definition at line 1649 of file transformations.py.
| list semanticmodel::transformations::_NEXT_AXIS = [1, 2, 0, 1] |
Definition at line 1652 of file transformations.py.
| tuple semanticmodel::transformations::_TUPLE2AXES = dict((v, k) for k, v in _AXES2TUPLE.items()) |
Definition at line 1665 of file transformations.py.